Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms

2
votes
0answers
39 views

Find a convergent solution for $a$

Find a value for $a$ in which the following sum converges. $$a+a!+(a!)!+((a!)!)!+\cdots$$ I know that there are no solutions if you only look at $a\in \Bbb{R}$, but are there any solutions if you ...
-1
votes
1answer
36 views

Find the natural number $n>2$ such that $\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!}$ [on hold]

I'm unsure how I'm supposed to solve the equation: $$\frac{n!}{(n-1)!} + \frac{n!}{3!(n-3)!} = 2\frac{n!}{2!(n-2)!} $$ given that $n>2.$
1
vote
3answers
80 views

How do you simplify $n!-(n-1)!$ [on hold]

I'm unsure how to simplify the expression $n!-(n-1)!$. Working as well as the final answer would be preferable.
0
votes
1answer
26 views

Can Stirling's approximation be used to obtain lower and upper bound for $\pi(x)$?

The Willan's formula is given as follows (taken from Ribenboim's Little book of bigger primes): $$ \pi(x)=\sum_{j=2}^{x}f(j) \text{ where } ...
1
vote
1answer
77 views

Solve the factorial equation $x! = c$

How to find the value of $x$ which its factorial for example equals to 100 ? $x! = 100 $ $x= ?$
2
votes
1answer
22 views

Reason of dividing to n! ( repetition ) on Permutations with Repetitions

I'm trying to figure out the reason of diving the number of permutations by the number of repetitions (in factorial). Shouldn't it be without the factorial? I don't get why are there is a factorial in ...
-1
votes
2answers
56 views

Prove by induction and recursion that $n!=n(n-1)(n-2)…(3)(2)(1). $

Prove by induction and recursion that $n!=n(n-1)(n-2)...(3)(2)(1). $ We can start with the definition of factorial with recursion: $$n!= \left\{\begin{align}1\quad \text{for}\quad ...
0
votes
1answer
50 views

Permutations excluding repeated characters

I'm working on a Free Code Camp problem - http://www.freecodecamp.com/challenges/bonfire-no-repeats-please The problem description is as follows - Return the number of total permutations of the ...
1
vote
1answer
52 views

Why is $\frac{1}{x} \sum_{n=1}^x \ln (n) \sim \ln(x) - \gamma$

I was playing with some functions and decided I wanted to see at which point the factorial of $x$ became bigger than $e^x$. I set them equal to each other and after doing some algebra I ended up with ...
0
votes
1answer
39 views

Finding a positive lower bound of hte sequence $\frac{\sqrt[n]{n!}}n$

I am given a sequence {(n'th root of n!)/n}. Can I show that the sequence is bounded below by a real no. which is greater than 0, by not calculating the limit of it....???thank you
2
votes
1answer
44 views

Infinitely many correct solutions to equation? [duplicate]

I conjecture that there are infinitely many correct solutions to this equation: (Where we are assuming $a,b \in \Bbb{N}$) $$a!+1=b^2$$ I chose to list the first three solutions below: $4!+1=5^2$ ...
0
votes
1answer
50 views

Solve limit using Stolz's theorem: $\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}$ [closed]

Solve this limit usinig Stolz's theorem. Any help?! $$\lim\limits_{n\rightarrow \infty} \frac{n}{\sqrt[n]{n!}}$$
1
vote
0answers
47 views

Why do the mathematicians stated $0!$ to be $1$? [duplicate]

My question is very simple, if just as we say $5! = 120, 4! = 24,$ how can we say that $0! = 1$? Why did the ancient mathematicians conventionally consider $0!$ to be $1$? Then there's coming lot of ...
3
votes
4answers
98 views

Why does $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ approximate $x!$ pretty well?

I was just messing around and trying out things in the desmos calculator and found that $\sqrt{3x} \left(\dfrac {x}{2} \right)^x$ is pretty close to $x!$ most of the time, here is a graph. Why does ...
11
votes
1answer
147 views

Proof that the factorial is nonelementary

Is there a proof that the factorial function $!:\mathbb N\to\mathbb N$ is nonelementary? If it were equal to an elementary function (call it $P(n)$), then it would extend the factorial function to ...
2
votes
1answer
66 views

How is this equation evaluated $\binom {n}2 = \frac{n^2}{2}$?

I would like to know how $\binom {n}2 = \dfrac{n^2}{2}$ works out while I'm reading a proof on this page. I have tried several ways, but I couldn't. i.e. we knew that combinatorics formula that ...
1
vote
1answer
65 views

Wilson's Theorem Factorial

I need to prove that $ (1 \cdot 3 \cdot 5 \dotsm 2009)^2 - 1 \equiv 0 \pmod{2011}$ By modular simplification, I need to prove that $(3 \cdot 5 \cdot 7 \dotsm 2009) \equiv 1 \pmod{2011}$ I know that ...
0
votes
4answers
630 views

The sum of the factors of 9! [closed]

The sum of the factors of 9! which are odd and of the form 3m+2(m is a natural number) is equal to $(A)40\hspace{1 cm} (B)42\hspace{1 cm}(C)46\hspace{1cm}(D)52$ I could not identify factors,i think ...
5
votes
1answer
206 views

Wilson's Theorem - Why only for primes? [closed]

Why is it true that Wilson's Theorem only holds for prime numbers? I read a proof of it, and it did not seem to cater to that aspect of the theorem.
-2
votes
1answer
32 views

A basic factorial question type [closed]

Hello could you show me a way that how to solve this kind of questions? a and b are natural numbers $60! = a6^b$ What is the biggest value of b?
21
votes
4answers
1k views

Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
1
vote
1answer
54 views

Use stirlings approximation to prove inequality.

I have come across this statement in a text on finite elements. I can give you the reference if that will be useful. The text mentions that the inequality follows from Stirling's formula. I can't ...
0
votes
1answer
29 views

Help evaluating a partial sum with factorials and binomial coefficients

I come from a CS background and had to contend with a problem similar to this one. Essentially, I want a general-case estimate on how many rolls I'd have to make to land on the same number twice with ...
3
votes
4answers
74 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
0
votes
0answers
29 views

How is zero factorial equals 1 [duplicate]

Simple question enough: Why is 0!=1 ? Had they chose it or proved it? Moreover is there any application of zero factorial?
0
votes
4answers
60 views

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$

Find the Limit of the sequence $\frac{(2^n)(n!)}{(2n+1)!}$ I'm not sure how to approach this problem. I tried the squeeze method, but could not figure it out.
0
votes
4answers
120 views

How many $0$s does the number $30!$ have? [duplicate]

I want to find out the number of $0$s in the number $30!$, what should I do? Is there any trick that would work for a general question of this type, like number of $0$s in $50!$ ?
0
votes
2answers
52 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
0
votes
1answer
48 views

Formula for reciprocal of a factorial

I was looking at some code here - https://www.codechef.com/viewsolution/6075682 when I came across this statement to calculate reciprocal of a factorial- ...
2
votes
1answer
23 views

How to find the next higher combination out of a fixed group of digits?

I have a group of contiguous digits ordered from smallest to highest: 1234. I want a formula (in case it exists) to find the next closer higher combination of the same digits. In this example the next ...
2
votes
0answers
46 views

Simplify ratio in factorial form [closed]

Is it possible to simplify the following ratio $$\frac{(\sum_{i=1}^{N}x_i)!}{(\prod_{i=1}^{N}x_i!)}$$ where $x_i\in\{0,1,2,..,M\}$
1
vote
0answers
32 views

Product of complex numbers $m+in$ with $0 < m,n \leq N$

I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case: $$ \log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$ For the ...
39
votes
3answers
695 views

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid ...
2
votes
2answers
183 views

How to sum factorials: $(n+1)! + n!$

How can the sum of factorials $(n+1)!+n!$ be simplified?
-1
votes
3answers
97 views

How to prove this inequality $(\frac{n+1}{e})^{n} < n! < e(\frac{n+1}{e})^{n+1}$? [closed]

$\Bigl(\frac{n+1}{e}\Bigr)^{n} < n! < e\Bigl(\cfrac{n+1}{e}\Bigr)^{n+1}$
3
votes
3answers
102 views

How many of the numbers in $A=\{1!,2!,…,2015!\}$ are square numbers?

Problem How many of the numbers in $A=\{1!,2!,...,2015!\}$ are square numbers? My thoughts I have no idea where to begin. I see no immediate connection between a factorial and a possible square. ...
2
votes
2answers
50 views

Wilson's theorem

According to Wilson's theorem, when p is prime (p-1)! mod p = -1 or p-1 What's the remainder in cases of (p-2)! mod p or ...
2
votes
1answer
76 views

Find the sum of the infinite series [1+(5/1!)+(8/2!)+(11/3!)+…]

Excluding the 1st term of the series 1, if we start from the 2nd term-($\frac{5}{1!}$), I can locate that the numerators are in A.P with common difference 3, & 1st term 5. Whereas the ...
2
votes
2answers
55 views

Is it possible to determine the number divisors of n! especially for large n?

I read this paper by P. Erdos, page 2. I didn't understand it. How do I determine the number divisors of $n!$ ? I'd like an example application, for example if I want to determine the number divisors ...
0
votes
0answers
34 views

Math Factorials. Simplifying by distrubution. I am confused.

Say we are working with statistics and factorials. In the proof of ... $$\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!(n-[n-r])!}$$ How is $(n-r)!(n-[n-r])!$ supposed to distribute to the simplified ...
3
votes
4answers
2k views

Calculate 2000! (mod 2003)

Calculate 2000! (mod 2003) This can easily be solved by programming but is there a way to solve it, possibly with knowledge about finite fields? (2003 is a prime number, so mod(2003) is a finite ...
2
votes
1answer
45 views

Number of digits in the number $N=(1.6 \times 10^{32})!$

I am trying to find the number of digits in $$N=(1.6 \times 10^{32})!$$ where ! denotes Factorial. I have no idea how to proceed, please help me.
3
votes
0answers
50 views

The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...
1
vote
3answers
64 views

Help with proof of $(n+1)^n > n! 2^n$

I have already managed to prove it using induction and Bernoulli's inequality but I wonder if there is another way. My proof goes like this: (This is my first time using MathJax, so I apologize for ...
2
votes
3answers
220 views

What function approximates the growth in length of factorial?

The factorial function grows in length in digits faster and faster. For example, early on it is multiplied by tens, so grows one or two digits each time. Then in the hundreds it grows two or three ...
0
votes
1answer
66 views

Sum of primes at minimal $\gt t!$

$$2+3+5+17+97+599\cdots a_t \gt t!$$ What does that mean? Well it is a sum that follows specific rules. For one, the number of terms in the sequence is $t$. Similarly, $a_t$ represents the $t$'th ...
0
votes
0answers
40 views

Factorial-Like Symbol for polynomials?

Is there a symbol similar to the factorial for polynomials? Like if I say $4!=4\times 3\times 2\times 1$ What is the equivalent operation such that Operation: $(x)(4)=x^4+x^3+x^2+x$ Where $x$ is ...
0
votes
1answer
29 views

Comparison between exponential and factorial results

I'm developing an algorithm to compare if the result of $n!$ is bigger than $k^m$, but I have problems with big integers, then I need to know if there's some property that I can use to do this without ...
3
votes
3answers
69 views

Number of primes from $n!+1$ to $n!+n$

Why aren't there any primes between $n!+1$ and $n!+n$ for all $n>1$? This question was on AHSME 1969 #23, but the question is trivial because it's multiple choice. However, I have no idea how to ...
3
votes
9answers
170 views

Why is $0! = 1$ the same as $1! = 1$? [duplicate]

I want to ask why is $$0! = 1$$ the same as $$1! = 1.$$ As a student I was lost and when I tried to ask the question the teacher said this will be done in complex analysis. I know here I will ...